Trends in electronic structures and $s_{\pm}$-wave pairing for the rare-earth series in bilayer nickelate superconductor $R_ 3$Ni$_2$O$_7$

The recent discovery of pressure-induced superconductivity in the bilayer La$_3$Ni$_2$O$_7$ (LNO) has opened a new platform for the study of unconventional superconductors. In this publication, we investigate theoretically the whole family of bilayer 327-type nickelates $R_3$Ni$_2$O$_7$ ($R$ = Rare-earth elements) under pressure. From La to Lu, the lattice constants and volume decrease, leading to enhanced in-plane and out-of-plane hoppings, resulting in an effectively reduced electronic correlation $U/W$. Furthermore, the Ni's $t_{2g}$ states shift away from the $e_g$ states, while the crystal-field splitting between $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ is almost unchanged. In addition, six candidates were found to become stable in the Fmmm phase, with increasing values of critical pressure as the atomic number increased. Similar to the case of LNO, the $s_{\pm}$-wave pairing tendency dominates in all candidates, due to the nesting between the {\bf M}=$(\pi,\pi)$ and the {\bf X}=$(\pi,0)$ and {\bf Y}=$(0,\pi)$ points in the Brillouin zone. Then, $T_c$ is expected to decrease as the radius of rare-earth (RE) ions decreases. Our results suggest that LNO is already the "optimal" candidate, with Ce a close competitor, among the whole RE bilayer nickelates, and to increase $T_c$ we suggest to grow on special substrates with larger in-plane lattice spacings.